Differentiation

Functions are generally classified in two categories under Calculus, namely

(i) Linear function

(ii) Non-linear functions

A linear function varies with a constant rate through its domain. Therefore, the overall rate of change of the function is same as the rate of change of a function at any point.

However, the rate of change of function varies from point to point in case of non-linear functions. The nature of variation is based on the nature of the function.

The rate of change of a function at a particular point is defined as a derivative of that particular function.

What is differentiation?

Differentiation can be defined as a derivative of a function regarding the independent variable and can be applied to measure the function per unit change in the independent variable.

Let y = f(x) be a function of x. Then, the rate of change of “y” per unit change in “x” is given by

dy / dx

If the function f(x) undergoes an infinitesimal change of h near to any point x, then the derivative of the function is defined as

\(\lim\limits_{h \to 0} \frac{f(x+h) – f(x)}{h}\)

When a function is denoted as y=f(x), the derivative is indicated by the following notations.

  1. D(y) or D[f(x)] is called Euler’s notation.
  2. dy/dx is called Leibniz’s notation.
  3. F’(x) is called as Lagrange’s notation.

Differentiation is the process of determining the derivative of a function at any point.

Differentiation Formulas

Some of the important Differentiation formulas in differentiation are as follows.

  1. If f(x) = tan (x), then f'(x) = sec2x
  2. If f(x) = cos (x), then f'(x) = -sin x
  3. If f(x) = sin (x), then f'(x) = cos x
  4. If f(x) = ln(x), then f'(x) = 1/x
  5. If f(x) = \(e^{x}\), then f'(x) = \(e^{x}\)
  6. If f(x) = \(x^{n}\), where n is any fraction or integer, then f'(x) = \(nx^{n-1}\)
  7. If f(x) = k, where k is a constant, then f'(x) = 0

Differentiation Rules

Some of the basic differentiation rules that need to be followed are as follows.

(i) Sum or Difference Rule

If the function is sum or difference of two functions, the derivative of the functions is the sum or difference of the individual functions, i.e.,

If f(x)=u(x)±v(x)

then, f'(x)=u'(x)±v'(x)

(ii) Product Rule

If the function f(x) is product of two functions u(x) and v(x), the derivative of the function is,

If \(f(x) = u(x) \times v(x)\)

then, \(\mathbf { f'(x) = u'(x) \times v(x) + u(x) \times v'(x)}\)

(iii) Quotient rule

If the function f(x) is in the form of two functions [u(x)]/[v(x)], the derivative of the function is

If, \(f(x) = \frac{u(x)}{v(x)}\)

then, \(\large \mathbf { f'(x) = \frac{u'(x) \times v(x) – u(x) \times v'(x)}{(v(x))^{2}}}\)

Chain Rule-

If a function y = f(x) = g(u) and if u = h(x), then,

\(\large \mathbf{\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{\mathrm{d} y}{\mathrm{d} u} \times \frac{\mathrm{d}u }{\mathrm{d} x}}\)<

This plays a major role in the method of substitution that helps to perform differentiation of composite functions.

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Practise This Question

If Φ(x)=log(sin x)x,xn π, nϵ I and 313log(sin x)xdx=Φ(k)Φ(1) then the value of k is